My two cents is that Paul was referring to the inflated reputations that some of the apostles were getting among certain portions of the community, and the partisan spirit that was occurring as everybody tried to pump up their own favorite apostle, like the Jews did for their favorite rabbis or football fans do for their team today. He has to deal harshly with this issue in Corinth (1 Cor. 1:10-15 and elsewhere in the letter). He was speaking not of Peter's authority, but of the absurdity of these popularity contests.

Hi,
Perhaps there is some historical dispute related to it, but Peter also says roughly the same thing:
Acts 10:34-35 And Peter opened his mouth and said: "Truly I perceive that God shows no partiality, but in every nation any one who fears him and does what is right is acceptable to him.

My guess is that when they talk about "partiality" or "personal favortism" they are referring to God's judgement of all men. Peter for example needed God's salvation in the same way that every Christian does. God didn't give him any preferential treatment.

2:6 seemed to be some thing: That is, they seemed to be of some authority because they had walked with the Lord and were present at the Transfiguration. They seemed to be of such in the eyes of those false brethren because those who seem to be something seem to be something to fleshly human beings. (Peter Lombard)

2:8 Peter to the circumcision and Paul uncircumcision: For even this distinction makes no difference, since it was a dispensation of divine grace, except that the Jews, who by the ancient custom of the law were not allowed to associate with Gentiles because they had for long been separated from them by the law, were admitted to the faith in Christ through the blessed Peter, while the Gentiles would similarly be conducted to faith through blessed Paul. But afterwards the shared character of the faith brought them together as one. Indeed, they all worked as one, as is clearly shown to be the case by the facts themselves. In fact, even Paul called as many of the Jews as he could to faith, as he did in Corinth (Acts 18:8). And it is clear in many places that he disputed with the Jews in the synagogues. Peter also went to the Gentiles as often as he could and did not disdain to do so as with Cornelius. (Theodore of Mopsuestia)

2:9 seemed to be pillars: Or sustainer of the Church. They were held in higher esteem than the rest of the apostles because Christ always admitted them into his more private moments. (St. Bruno)

2:11 I withstood him to his face: Paul must be admired because he believed that truth must be honored before everything, and for it he did not hesitate to oppose the most excellent and distinguished of the Apostles to his face. But Peter must be admired because, granted that he appeared to be convicted, nevertheless he remained quiet, bearing it all with silence. Though he was capable of asserting his primacy on the basis of many considerations, he treated his own affairs as of no importance and thought that people should honor truth above all. But their agreement in the time that followed has demonstrated that their dispute did not cause any division. (Theodore of Mopsuestia)

Because he was to be blamed. It may be asked whether Peter was really blameworthy and was actually blamed by Paul. For many years there was a sharp dispute on this point between St. Jerome and St. Augustine, as may be seen in their epistles. Jerome, Chrysostom, Theophylact, Baronius answer in the negative, and hold that the rebuke was only theatrical. They argue that Peter, who had lawfully followed the Jewish customs at Jerusalem among Jews, lived as a Gentile among Gentiles at Antioch; when, however, the Jews arrived who had been sent to Antioch from Jerusalem by James, he withdrew from the Gentiles in favor of the Jews, lest he should offend those who had been the earliest to receive the faith (see ver. 9), and also that he might at the same time give Paul, the Apostle of the Gentiles, an opportunity of rebuking him, that by yielding he might teach the Jews that the time for Judaising was past. On the other side St. Augustine maintains that Peter was really blameworthy, and was blamed by Paul, as the record distinctly declares. (Cornelius a Lapide)